1 / 15

Computational Geometry The art of finding algorithms for solving geometrical problems

Computational Geometry The art of finding algorithms for solving geometrical problems. Literature: M. De Berg et al: Computational Geometry, Springer, 2000. H. Edelsbruner: Algorithms in Combinatorial Geometry, Springer, 1987. 1. Convex Hull. 1.1 Euclidean 2-dimensional space E 2

tannar
Download Presentation

Computational Geometry The art of finding algorithms for solving geometrical problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Computational GeometryThe art of finding algorithms for solving geometrical problems • Literature: • M. De Berg et al: Computational Geometry, Springer, 2000. • H. Edelsbruner: Algorithms in Combinatorial Geometry, Springer, 1987.

  2. 1. Convex Hull 1.1 Euclidean 2-dimensional space E2 • Real Vector Space V2(V,+,•); • Equations of lines in E2: a1x1 + a2x2 = b (eq. 1) X=A + (1-)B (eq. 2) l B (1-l) X A

  3. 1.2 Affine / Convex Combination Affine combination of points A and B A + mB, + m= 1 Convex combination of points A and B , m  0 A + mB, + m = 1, Generalization: Euclidean n-dim space En Affine combination:  1A1 +  2A2 + … +  kAk ,  1 +  2 + … +  k = 1 Convex combination:  1A1 +  2A2 + … +  kAk ,  1 +  2 + … +  k = 1,  1 ,…,  k 0

  4. 1.3 Affine / Convex Hull • Affine Hull of a finite set of points A1 ,…, Ak  1A1 +  2A2 + … +  kAk :  1 +  2 + … +  k = 1 • Convex Hull of a finite set of points A1 ,…, Ak  1A1 +  2A2 + … +  kAk :  1 +  2 + … +  k = 1,  i 0 Affine (Convex) Hull of a set S, notation Aff (S) (Conv (S)), is the set of all affine (convex) combinations of finite subsets of S.

  5. Examlpes • Line AB is the affine hull of A and B. • Plane ABC is the affine hull of affinely independent points A, B and C. • ... • Segment [A,B]. - is the set of points on AB which are between A and B, i.e. the set of convex combinations of A and B. • Triangle, ...

  6. 1.4 Exercises Exercise 1 Prove that if a point B belongs to the affine hull Aff (A1, A2, …, Ak ) of points A1, A2,…, Ak , then: Aff (A1, A2,…, Ak) = Aff (B,A1, A2,…, Ak).

  7. 1.4 Exercises (cont.) Exercise 2 • Prove that the affine hull Aff (A1, A2, …, Ak ) of points A1, A2,…, Ak contains the line AB with each pair of its points A,B. • Moreover, prove that Aff (A1, A2, …, Ak ) is the smallest set containing {A1, A2, …, A k } and having this property. [This property therefore may serve as a definition of affine sets. ] (Hint: proof by induction.)

  8. 1.4 Exercises (cont.) Exercise 3 Prove that Aff (A1,, A2,…, Ak ) is independentofthe transformation of coordinates. Definition: Affine transformation of coordinates: Matrix multiplication : X -> X• Mnn Matrix translation: X-> X + O’

  9. 1.4 Exercises (cont.) Reformulate exercises 1-3 by substituting: • Aff (A1, A2,…, Ak ) with Conv (A1, A2,…, Ak ) • line AB with segment [A,B]. • Definition: Convex set is a set which contains the segment [A,B] with each pair of its elements A and B. Exercise 1’-3’

  10. 1.4 Exercises (cont.) If a convex set S contains the vertices A1, A2,…, Ak of a polygon P=A1A2…Ak , it contains the polygon P. (Hint: Interior point property). Exercise 4

  11. 1.4 Exercises (cont.) Exercise 5-5’ Prove that Aff (S) (Conv ( S)) is the smallest affine (convex) set containing S, i. e. the smallest set X which contains the line AB (segment [AB]) with each pair of points A, B  X. Alternatively: Definition: Convex Hull of a set of points S, notation Conv (S ) is the smallest convex set containing S.

  12. 1.4 Exercises (cont.) Prove that: Aff (A1, A2,…, Ak) = Aff (A1,, Aff (A2,…, Ak)). Conv (A1, A2,…, Ak) = Conv (A1,, Conv (A2,…, Ak)). Exercise 6-6’

  13. 1.4 Exercises (cont.) Definiton: A set S of points is said to be affinely (convex) independent if no point of S is an affine combination of the others. Affine (convex) basisBS of a set S is an affinely (convex) independent subset of S such that every point in S is an affine (convex) combination of points from BS. A set A1, A2,…, Ak is affinely independent if and only if the set of vectors A1A2,…, A1Akis linearly independent. Exercise 7

  14. Therefore... Every set in En has an affine basis of k n+1 points. (easy to prove). It has also a convex basis (not easy to prove), which is not necessarily finite. A number of points in an affine basis of a set S is constant and said to be the dimension of S.

  15. Theorem If S is a finite set of points, then Conv (S) is a polygon with the vertices in S. Proof: 1. There is a convex polygon PS with the vertices in S. (for example, Algorithm1) 2. PS= Conv (S).

More Related